Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Efficient Memory Gradient Method for Extreme M-Eigenvalues of Elastic type Tensors

Zhuolin Du, Yisheng Song

TL;DR

This work targets efficient computation of extreme $M$-eigenvalues for fourth-order hierarchically symmetric (elastic-type) tensors by recasting the problem as an unconstrained optimization with a shift parameter. A Memory Gradient Method (MGM) is proposed, featuring a k-step history-integrated update ${\bf d}_k=-\gamma_k g({\bf z}_k)+\frac{1}{N}\sum_{i=1}^N\beta_{k_i}{\bf d}_{k-i}$, Wolfe line search, and a rescaling step to enforce $\|{\bf x}\|=\|{\bf y}\|$, with rigorous global convergence guaranteed via descent properties and Zoutendijk-type arguments. Theoretical results show bounded iterates, Lipschitz continuity of the gradient, and convergence to a stationary point or $\liminf\|g({\bf z}_k)\|=0$. Numerical experiments demonstrate that MGM, particularly MGM-1 with $N=3$, outperforms existing methods (SIPM and SS-HOPM) in both iteration count and runtime across diverse tensor instances and dimensions, highlighting its practical efficiency for targeted eigenvalue extraction in elasticity and quantum contexts.

Abstract

M-eigenvalues of fourth order hierarchically symmetric tensors play a significant role in nonlinear elastic material analysis and quantum entanglement problems. This paper focuses on computing extreme M-eigenvalues for such tensors. To achieve this, we first reformulate the M-eigenvalue problem as a sequence of unconstrained optimization problems by introducing a shift parameter. Subsequently, we develop a memory gradient method specifically designed to approximate these extreme M-eigenvalues. Under this framework, we establish the global convergence of the proposed method. Finally, comprehensive numerical experiments demonstrate the efficacy and stability of our approach.

An Efficient Memory Gradient Method for Extreme M-Eigenvalues of Elastic type Tensors

TL;DR

This work targets efficient computation of extreme -eigenvalues for fourth-order hierarchically symmetric (elastic-type) tensors by recasting the problem as an unconstrained optimization with a shift parameter. A Memory Gradient Method (MGM) is proposed, featuring a k-step history-integrated update , Wolfe line search, and a rescaling step to enforce , with rigorous global convergence guaranteed via descent properties and Zoutendijk-type arguments. Theoretical results show bounded iterates, Lipschitz continuity of the gradient, and convergence to a stationary point or . Numerical experiments demonstrate that MGM, particularly MGM-1 with , outperforms existing methods (SIPM and SS-HOPM) in both iteration count and runtime across diverse tensor instances and dimensions, highlighting its practical efficiency for targeted eigenvalue extraction in elasticity and quantum contexts.

Abstract

M-eigenvalues of fourth order hierarchically symmetric tensors play a significant role in nonlinear elastic material analysis and quantum entanglement problems. This paper focuses on computing extreme M-eigenvalues for such tensors. To achieve this, we first reformulate the M-eigenvalue problem as a sequence of unconstrained optimization problems by introducing a shift parameter. Subsequently, we develop a memory gradient method specifically designed to approximate these extreme M-eigenvalues. Under this framework, we establish the global convergence of the proposed method. Finally, comprehensive numerical experiments demonstrate the efficacy and stability of our approach.
Paper Structure (6 sections, 8 theorems, 85 equations, 3 figures, 2 tables)

This paper contains 6 sections, 8 theorems, 85 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Unconstrained framework for M-eigenvalues of elastic type tensors
  3. A memory gradient method and its convergence
  4. Numerical results
  5. Analysis of $\gamma_k$ and $N$
  6. Analysis of all compared algorithms

Key Result

Theorem 2.1

Let $\mathcal{A}\in \mathbb{H}^{m\times n\times m\times n}$ be a nonzero tensor. Assume that $\lambda^\ast$ is the largest M-eigenvalue of $\mathcal{A}$. Then we have the following results (1) The problem 2.1 has a global minimum. (2) Let ${\bf x}\in\mathbb{R}^m\setminus\{\bf0\}$ and ${\bf y}\in\mat

Figures (3)

  • Figure 2: Performance profiles for MGM-1 with different $N$ values.
  • Figure 4: Full performance of both MGM-1, MGM-2, SIPM and SS-HOPM in terms of the averaged values and standard derivation of 10 random tests.
  • Figure : (a) Iterations

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Lemma 3.1
  • ...and 9 more